A Simple Family of Analytical Trumpet Slices of the Schwarzschild Spacetime

Abstract

We describe a simple family of analytical coordinate systems for the Schwarzschild spacetime. The coordinates penetrate the horizon smoothly and are spatially isotropic. Spatial slices of constant coordinate time \(t\) feature a trumpet geometry with an asymptotically cylindrical end inside the horizon at a prescribed areal radius \(R_0\) (with \(0<R_{0}\leq M\)) that serves as the free parameter for the family. The slices also have an asymptotically flat end at spatial infinity. In the limit \(R_{0}=0\) the spatial slices lose their trumpet geometry and become flat -- in this limit, our coordinates reduce to Painlev\'e-Gullstrand coordinates.